1. Lecture 10: Observers and Kalman Filters
CS 344R: Robotics
Benjamin Kuipers

2. Stochastic Models of an Uncertain World
Actions are uncertain.
Observations are uncertain.
i ~ N(0,i) are random variables

3. Observers The state x is unobservable.
The sense vector y provides noisy information about x.
An observer is a process that uses sensory history to estimate x.
Then a control law can be written

4. Kalman Filter: Optimal Observer

5. Estimates and Uncertainty
Conditional probability density function

6. Gaussian (Normal) Distribution
Completely described by N(,)
Mean 
Standard deviation , variance  2

7. The Central Limit Theorem
The sum of many random variables
with the same mean, but
with arbitrary conditional density functions,
converges to a Gaussian density function.
If a model omits many small unmodeled effects, then the resulting error should converge to a Gaussian density function.

8. Estimating a Value Suppose there is a constant value x.
Distance to wall; angle to wall; etc.
At time t1, observe value z1 with variance
The optimal estimate is with variance

9. A Second Observation
At time t2, observe value z2 with variance

10. Merged Evidence

11. Update Mean and Variance
Weighted average of estimates.
The weights come from the variances.
Smaller variance = more certainty

12. From Weighted Average to Predictor-Corrector
This version can be applied “recursively”.

13. Predictor-Corrector Update best estimate given new data
Update variance:

14. Static to Dynamic
Now suppose x changes according to

15. Dynamic Prediction At t2 we know
At t3 after the change, before an observation.
Next, we correct this prediction with the observation at time t3.

16. Dynamic Correction At time t3 we observe z3 with variance
Combine prediction with observation.

17. Qualitative Properties
Suppose measurement noise is large.
Then K(t3) approaches 0, and the measurement will be mostly ignored.
Suppose prediction noise is large.
Then K(t3) approaches 1, and the measurement will dominate the estimate.

18. Kalman Filter Takes a stream of observations, and a dynamical model.
At each step, a weighted average between
prediction from the dynamical model
correction from the observation.
The Kalman gain K(t) is the weighting,
based on the variances and
With time, K(t) and tend to stabilize.

19. Simplifications We have only discussed a one-dimensional system.
Most applications are higher dimensional.
We have assumed the state variable is observable.
In general, sense data give indirect evidence.
We will discuss the more complex case next.